In this paper, we have discussed special solutions to a trio of bi-Riccati delay-differential equations. The equations we study are distinguished by their singularity and (elliptic) solution structures but are also shown to arise as simple reductions of known integrable semi-discrete equations. In this sense, our equations are analogs of particular instances of the QRT map [1]}. The Cauchy problem for ordinary difference (or ordinary differential) equations prohibits the general solution from containing hierarchies of soliton-type solutions. However, the solution structure of delay-differential equations is much richer than that of equations with only shifts or only derivatives and may, for particular equations, contain such hierarchies. We observe this phenomenon in two of the equations under consideration by means of the Hirota bilinear method.
